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ORIGINAL PAPER
The impact of geological uncertainty on primary productionfrom a fluvial reservoir
Mohammad Koneshloo1 • Saman A. Aryana1,2 • Xiaoni Hu3
Received: 4 July 2017 / Published online: 21 April 2018� The Author(s) 2018
AbstractDeposition of fluvial sandbodies is controlled mainly by characteristics of the system, such as the rate of avulsion and
aggradation of the fluvial channels and their geometry. The impact and the interaction of these parameters have not
received adequate attention. In this paper, the impact of geological uncertainty resulting from the interpretation of the
fluvial geometry, maximum depth of channels, and their avulsion rates on primary production is studied for fluvial
reservoirs. Several meandering reservoirs were generated using a process-mimicking package by varying several con-
trolling factors. Simulation results indicate that geometrical parameters of the fluvial channels impact cumulative pro-
duction during primary production more significantly than their avulsion rate. The most significant factor appears to be the
maximum depth of fluvial channels. The overall net-to-gross ratio is closely correlated with the cumulative oil production
of the field, but cumulative production values for individual wells do not appear to be correlated with the local net-to-gross
ratio calculated in the vicinity of each well. Connectedness of the sandbodies to each well, defined based on the minimum
time-of-flight from each block to the well, appears to be a more reliable indicator of well-scale production.
Keywords Geological uncertainty evaluation � Fluvial reservoir modeling � Process-mimicking simulation �Geometry of fluvial channels
1 Introduction
Fluvial reservoirs are heterogeneous at a multitude of
scales. High-permeability contrasts between mud and sand
deposits make the connectivity of sandbodies the main
controlling factor for reservoir quality. The internal struc-
tures of sediments on a small scale (core and pore scales)
and stratigraphic correlation of fluvial formations on a
large scale have received more attention than the archi-
tecture of depositions, the geometry of channels, and
conserved sandbodies, which are more important for the
geological modeling of a reservoir (Gibling 2006). The
sources of geological uncertainty are classified by Mas-
sonnat (2000) in a six-level hierarchical chart: data
acquisition, interpretation of data, geological concept
elaboration, the spatial distribution of geological features,
choice of the input parameters, and generation of
equiprobable realizations. Despite the importance of these
factors, geological models are assumed fixed or predeter-
mined in many routine industrial appraisal projects, and
practitioners often prefer to focus only on one fine geo-
logical model and to analyze only the impact of dynamic
parameters. Laure and Hovadik (2008) state two reasons
for this lack of attention; first, it is believed that reservoir
architecture is a less important factor in oil production than
other petrophysical and fluid properties or the operational
parameters of a reservoir. Second, generating realistic
subsurface models is a difficult and time-consuming task.
In fact, the first reason may be true only if the lithological
facies demonstrate similar patterns or exhibit similar con-
nectivity despite their different patterns. The difficulty of
subsurface modeling resides in its uncertain nature coming
from lack of subsurface data which lead to various possible
Edited by Jie Hao
& Mohammad Koneshloo
1 Department of Petroleum Engineering, University of
Wyoming, Laramie, WY 87061, USA
2 Department of Chemical Engineering, University of
Wyoming, Laramie, WY 87061, USA
3 Department of Geological Sciences, University of Wyoming,
Laramie, WY 87061, USA
123
Petroleum Science (2018) 15:270–288https://doi.org/10.1007/s12182-018-0229-y(0123456789().,-volV)(0123456789().,-volV)
interpretations of the geological settings, where the sedi-
mentological and structural factors are thought to be the
major sources of uncertainty. A specific geological model,
however, can never capture the full range of geological
uncertainty of a reservoir (Caumon and Journel 2005).
Fluvial reservoirs and especially meandering deposits
appear to demonstrate the importance of reservoir archi-
tecture in project appraisal. The architecture of the mean-
dering deposits, formed by multi-story channel deposits
separated by floodplain mud, is complex as a result of
lateral migration and avulsion. The sand fraction is known
as the dominant factor in evaluation of a fluvial reservoir,
while the geometry of channels, avulsion rate, and direc-
tion of point-bar migration are also significant (Jones et al.
2009). These factors also determine the connectivity of
sandbodies (high permeable regions), which is a dominant
factor influencing fluid flow and shaping preferential flow
pathways in reservoirs (Pranter and Sommer 2011). All of
these characteristics are controlled by the geological
parameters of the system. Correct estimation of sandbodies
and their topological relationships are key elements in
understanding these reservoirs. This paper aims to
demonstrate the impact of the geometry of fluvial channels
and avulsion rates on the quality of a fluvial reservoir.
Numerical reservoir modeling is a commonly used tool
for analyzing the impact of geological features on pro-
duction. Most stochastic reservoir modeling methods rely
on spatial correlation and well data to generate represen-
tative models. In such a framework, it is difficult to
incorporate physical laws of sedimentation and prevent
geologically incorrect placement of sandbodies (Tye 2013).
Methods such as object-based methods (Pyrcz and Deutsch
2014) and multiple-point geostatistics (Strebelle 2002;
Rezaee et al. 2013) rely on predefined patterns or objects,
which are the consequence of sedimentation under a
defined set of geological factors. Process-based simulators
are tools developed to mimic the sedimentation process and
to evaluate the impact of geological factors directly (De-
viese 2010; Hajek and Wolinsky 2012). Most process-
based simulators incorporate a simplified equation of flow
to mimic processes governing deposition of sediments
(Cojan et al. 2005). In this study, subsurface models are
generated using FLUMY, a process-mimicking software
for meandering systems (Koneshloo et al. 2017).
This work focuses on levels three and four of the
Massonnat (2000) hierarchical classification of the sources
of uncertainty: elaboration of the geological model and
explanation of the main factors controlling the distribution
of the geological features in a meandering reservoir.
According to Tye (2013), the most important factors con-
trolling the static connectivity of sedimentary sandbodies
are channel widths and net-to-gross ratio (NGR) values.
Although NGR values calculated using well data (1-D) are
generally used as acceptable approximations of the reser-
voir NGR, the interpretation of the shape and lateral
extension of sedimentary features solely based on vertical
data is rather difficult. In this work, several models are
generated to study the impact of uncertainty inherent in
subsurface model conceptualization and the controlling
factors. The following sections describe the geometry of
channels, present several empirical equations used to
explain the geometry of channels, and discuss ambiguities
in their application. Petrophysical models are generated,
and oil production from each model is simulated. Field-
level and well-level results are compared, and the impact of
various geological factors on cumulative primary produc-
tion is analyzed.
2 Materials and methods
In geometrical modeling of a reservoir’s features, we are
often faced with a scarcity of data. Three-dimensional
seismic data are often used to generate pseudo-images of
the subsurface. Several studies show that incised valleys,
channel belts, large channels (with a thickness above the
seismic vertical resolution), and abandoned channels filled
with coaly horizons are seismically detectable (Alsouki
et al. 2014; Zhang et al. 2015). In the absence of seismic
data, outcrops and modern deposition systems are impor-
tant sources of information related to sedimentary features.
For a given reservoir, architecture dictates the spatial dis-
tribution of reservoir properties within the reservoir. The
distinction between all sedimentological features and reli-
able estimation of their geometrical parameters are not
always possible. Although core and log data have good
vertical resolutions, lateral continuity cannot be extracted
directly from such data.
Miall (2014) illustrates the uncertainty inherent in geo-
logical modeling using three interpreted sections of zone
one of the Travis Peak Formation. In this study, three
different empirical equations are used to estimate channel
widths based on their depths (Bridge and Mackey 2009;
Leeder 1973; Williams 1986). Due to differences in
assumptions, the same logs result in three completely dif-
ferent representations of the subsurface in terms of the
connectivity of sandbodies. Additional challenges include
synonymous use of terms (e.g., meander belt and channel)
and inconsistent estimation of influential parameters (e.g.,
width of fluvial channels may be estimated based on either
maximum bankfull depth, or mean bankfull depth of the
channels, or thickness of sandbodies). Finally, these vari-
ables must often be inferred based on limited one-dimen-
sional data. Table 1 shows the most commonly used
empirical equations for the estimation of lateral extension
Petroleum Science (2018) 15:270–288 271
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(width) of fluvial channels based on their depths. The
parameters are described schematically in Fig. 1.
As pointed out by Miall (2014), the rate and style of
avulsion are both important factors affecting mid-range
heterogeneities and architecture of fluvial depositions,
which are the focus of the present paper. Avulsion occurs at
two distinct scales: regional and local. The impact of
avulsion rate and its style is investigated by varying
aggradation rate and regional avulsion rate.
The local avulsion rate is kept constant due to its rela-
tively limited impact on the overall architecture. In the
present paper, it is assumed that sedimentation is controlled
by only a few important factors, namely avulsion fre-
quency, floodplain aggradation rate, and the ratio of
Table 1 Summary of empirical equations for the estimation of the geometry of fluvial systems
Model Parameters How to measure Limitations
Bridge and Tye (2000)
Wc ¼ 8:8� H1:88mean
Hmax: Bankfull
channel depth,
m
Hmean: Mean
bankfull
channel depth,
m
Wc: Channel
width, m
Based on the thickness of sandbodies defined as a single
and untruncated story
Difficult to find single and
untruncated stories
Leeder (1973)
Wc ¼ 6:8� H1:54max
Wc: Bankfull
channel width,
m
Hmax: Bankfull
channel depth,
m
Bankfull depth: max vertical distance between point bar
and channel thalweg
Upward fining trend is not
representative of point-bar
thickness due to erosion
Williams (1986)
Wc ¼ 21:3� H1:45mean
Hmean: Mean
bankfull
channel depth,
m
Wc: Bankfull
channel width,
m
Bankfull mean depth computed based on three cross
section in field and topographic maps
No explanation on how these
parameters can be measured in
a fluvial reservoir
Collinson (1977)
Wmb ¼ 64:6� H1:54max
Wmb: Meander
belt width
Hmax: Bankfull
channel depth,
m
– Upward fining trend is not
representative of point-bar
thickness due to erosion
Lorenz et al. (1985)
Wc ¼ 6:8� H1:54max
Wmb ¼ 4:77�W1:01c
Wc: Bankfull
channel width,
m
Hmax: Bankfull
channel depth,
m
Wmb: Meander
belt width, m
Bankfull depth: Thickness of the point bars, each upward
fining trend is assumed as a preserved point bar
Upward fining trend is not
representative of point-bar
thickness due to erosion
Fielding and Crane (1987)
Hmax ¼ 0:55� th
Wcb ¼ 64:6� H1:54max
(channels allowed to
develop full meander
profile)
Hmax: Channel
depth, m
th: Sandstone-
body
thickness, m
Wcb: Channel
belt width
Identification of channel is based on their upward fining
profile, and th\ 2 m is considered as non-channel,
crevasse splays deposition
Estimation of channel depth is
difficult due to erosion and
accretion
The first three equations describe the relation between bankfull channel depth (maximum or mean) and the width of fluvial channels. The
remainder of the equations show empirical relations between width of meander or channel belt and channel bankfull depth (the geometrical
parameters are depicted in Fig. 1)
272 Petroleum Science (2018) 15:270–288
123
channel belt or of floodplain to the width in the stacking of
channel bodies.
Avulsion controls the reoccupancy of channel belts,
stacking and distribution of sandbodies and serves as a
primary factor in shaping the architecture of fluvial depo-
sitions (Jones and Schumm 1999; Miall 2014; Slingerland
and Smith 2004). Figure 2 shows the impact of the rate of
aggradation, bank migration, and avulsion on the shape and
connectivity of sandbodies. The focus of the present
research is on the role of avulsion frequency (Av), channel
bankfull depth (Hmax), and shape of channels (presented by
different relationships between channel width and water
depth) in the architecture of reservoirs.
2.1 Geological modeling
Process-based simulators generate complex models that
cannot be easily conditioned with the available data;
therefore, process-based simulators are rarely used directly
to generate subsurface models (Karssenberg et al. 2001).
Process-imitating models are simplified process-based
simulators that only consider the most significant process-
based factors, therefore requiring fewer parameters
(Koneshloo et al. 2017). Process-imitating software,
FLUMY, is used as a proxy for process-based simulators.
This package, developed at MINES ParisTech (Centre de
Geosciences in Fontainebleau, France), generates realistic
models for fluvial meandering systems (Deviese 2010;
Lopez et al. 2009), by tracking the evolution of meandering
channels. The list of the model’s parameters is provided in
Table 2. FLUMY makes use of a limited number of key
Fig. 1 Schematic demonstration of channel geometry and the relevant
geometrical parameters. The left panel shows a cross section of flow
and the geometrical parameters of the fluvial channel. In the right
panel, Ucayali River (Rio Ucayali) in Peru is used as an example to
show channel migration (1969 shown by solid black lines, and 2016
shown by orange). Point bars, crevasse splay, and two hypothetical
paths for regional (dark blue path) and local avulsion (purple path) are
schematically presented
Fig. 2 Role of geological factors controlling the connectivity and the shape of deposited sandbodies. (Reproduced with permission from Gibling
2006)
Petroleum Science (2018) 15:270–288 273
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parameters (fewer than what is needed for a process-based
model). The main controlling factors are regional avulsion
rate, aggradation rate, and general topography (slope of the
basin). In each iteration, FLUMY calculates migration of
the centerline; details are described in Cojan et al. (2005)
and Lopez et al. (2009). For each pixel, sediment will be
deposited according to its distance from the centerline. An
exponential function for distance from the centerline is
used to assign the grain size and the thickness of the sed-
iments during each iteration (Lauer and Parker 2008; Piz-
zuto et al. 2008).
In FLUMY, regional avulsion is defined as an avulsion
happening upstream of the domain and local avulsion
refers to a levee breach inside the domain. The frequency
of the local avulsion (levee breaches) and regional avul-
sion, and the aggradation rate of the floodplain used in this
study are specified Table 2. Other variables include the
equation used to relate channel width to channel depth,
avulsion periodicity (Av), and maximum channel depth
(Hmax).
A 3D model of a fluvial meandering system with
approximately 7 million cells (216 9 216 9 150) is sim-
ulated. This simulated field is 75 m thick, 3240 m wide,
and 3240 m long and is used as the reference model. Cell
dimensions are 0.5 m in the z-direction and 15 m in the x-
and y-directions. There are two possible sources of ran-
domness in FLUMY.
The first source is the randomness of the model’s
parameters, and the second source resides in the way the
algorithm has been designed to simulate the fluvial regime,
initial river path, and its new path caused by regional
avulsion. Parameters used in the generation of the reference
model are based on unit four of Schooner Field, as reported
by Deviese (2010). Lithological data are sampled along 16
vertical wells located approximately at centers of 80-acre
parcels in the reference model. Subsequently, fifty-four
geological models are generated using FLUMY, by varying
the maximum depth of the fluvial channel, the equation
used to estimate channel widths, and the avulsion rate.
Figure 3 shows the width of channels calculated using
the first three equations in Table 1. As shown in Fig. 4,
William’s formula generates a larger fluvial channel, and
Bridge and Mackey’s formula leads to a narrower channel.
To generate a 3D model of the facies, data are sampled
every 0.5 m vertically and lithotypes are grouped together
under three main facies: coarse sediments (SS), which are
common deposit bars in fluvial channels; crevasse splays
(CS), which are finer sands adjacent to main channel
Fig. 3 Relation between maximum depth of meandering channels and
their width
Table 2 Parameter set used in FLUMY to generate geological models
Parameter, unit Default value Value used for modeling
Channel bankfull width, m 100 Based on the relationship used and its
maximum depth
Maximum bankfull depth, m 1/10 of channel width 2.0; 2.5; 3.0; 3.3; 3.5; 3.7; 4.0
Mean bankfull depth 2/3 of maximum depth for parabolic
cross section
2/p of maximum depth for ellipse shape
section
Bank erodibility coefficient (based on channel
migration in m/s)
4 9 10-8 6.5 9 10-8
Regional avulsion, year Type: Periodic period = 2200 1 9 104; 4 9 104; 7 9 104
Local avulsion or levee breach, year Type: Periodic period = 600 Default
Probability of transition from CSI to CSII 0.9 Default
Aggradation, year Type: Periodic period = 100 Default
Thickness exponential decrease, m 1000 2000
Grain size exponential decrease, m 1000 2000
Levee width (multiple of the channel width) 6 Default
Slope along flow direction 1 9 10-3 Default
274 Petroleum Science (2018) 15:270–288
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deposits; and overbank deposits (OB), which are usually
floodplain muds. The preservation potential of each geo-
logical feature is a crucial point in fluvial reservoir mod-
eling. As shown in Figs. 4 and 5, attempts to use current
geomorphologic data to define the objects’ geometry and
collocation often lead to an incorrect subsurface model
(Pyrcz and Deutsch 2014). Preferential paths (highly per-
meable facies) are not sinusoidal channels, but a complex
Fig. 4 Cross section of the fluvial system generated based on a Eq. 1: Bridge and Tye (2000), b Eq. 2: Williams (1986), and c Eq. 3: Leeder
(1973)
Petroleum Science (2018) 15:270–288 275
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aggregation of lobe-shaped sediments (Fig. 5b). Sixty-
three static models were generated and primary oil pro-
duction simulated for each realization.
2.2 Petrophysical modeling
Porosity and permeability control the flow and fluid storage
capacity of a reservoir and are largely determined by
lithology and grain size, sorting, and consolidation (Slatt
2006). Porosity values are simulated independently for
each facies using sequential Gaussian simulation using
parameters shown in Table 3. A cookie-cutter approach
(Mao and Journel 1999), shown schematically in Fig. 6, is
used to generate the porosity model for each case. Per-
meability (k) is calculated based on the porosity value (u)and the type of facies (represented by their grain size, dlm),
using the Kozeny–Carman equation (Mao and Journel
1999). Tortuosity is assumed to be 2.5 (Mao and Journel
1999), and pore diameter (in lm) is defined as a function of
facies type. This work assumes that pore sizes are pro-
portional to grain sizes following a cubic packing form
where pore diameters are 47% of grain diameters. Then a
random noise with a mean of 0.0 and a standard deviation
of 0.5 is added to the logarithmic value of calculated k, as
k ¼ 1
180
u3
1� uð Þ2d2lm þ noise:
Figure 7 shows the histogram and scatter plot of
porosity and permeability values of the facies in the ref-
erence model. The mean and standard deviation of porosity
for each facies are based on Rocky Ridge field in North
Dakota (Hastings 1990) and Byrnes’s study on low-per-
meability sandstones in the Rocky Mountains (Byrnes
1997). To ensure reasonable simulation times, the gener-
ated static models are upscaled. Porosity and permeability
are upscaled using arithmetic and geometric averaging,
respectively (Jensen 1991). Figure 8 shows fine-scale and
upscaled reference model.
2.3 Reservoir modeling
Fluid properties are shown in Table 4. The reservoir is
assumed to be part of a larger field; therefore, the bound-
aries of the model are considered open boundaries. Relative
permeability relations are assumed to be sole functions of
local saturation (Ren et al. 2017). As the type of facies, and
their porosity and absolute permeability values vary within
the reservoir, the irreducible water and oil saturations vary
as well. Thus, three different rock regions are defined based
on the facies. Relative permeability and capillary pressure
relations are assigned to each facies (Fig. 9). Three dif-
ferent datasets are used for theses facies: sandstone, tight
sand, and shale rock reservoirs. The main reasoning behind
Fig. 5 a 3D view of the reference geological model (realization), b sediments having a permeability more than 1 mD
Table 3 Variogram models
used to generate porosity valuesParameters Facies
Sandstones (SS) Crevasse splays (CS) Overbank deposits (OB)
Nugget 1% 1% 1%
Type Spherical Spherical Spherical
Ranges, m 1000, 500, 10 500, 200, 5 750, 750, 7
Angles 90, 0, 0 (flow direction) 90, 0, 0 (flow direction) 0, 0, 0
Average 0.16 0.10 0.06
Standard deviation 0.025 0.025 0.020
276 Petroleum Science (2018) 15:270–288
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this choice is the similarity between the facies and their
lithology. For sandstones, the data from the Rocky Ridge
field in North Dakota are used (Hastings 1990). As cre-
vasse splays are often considered non-reservoir rock, only
few measurements of their relative permeability and cap-
illary pressure relations are available (e.g., Ramon and
Cross 1997). This can be explained by the fact that cre-
vasse splays facies are a small proportion of fluvial reser-
voirs. In addition, they represent low-quality reservoirs due
to the inclined mud drapes forming barriers to both vertical
and horizontal flows. In this study, the data from a tight
formation in Canada (Zhang et al. 2012) have been used for
crevasse splays.
Overbank groups (including wetlands, mud plugs, and
overbank) are assumed to be fine silt to clay, which are
similar to typical shale formations; therefore, for
overbanks, constitutive relations are based on published
experimental data from the Bakken reservoir’s organic-rich
Bakken shale (Nojabaei et al. 2014).
A streamline-based flow simulator, studioSL� from
StreamSim, is used for fluid flow simulation due to its low
computational cost relative to traditional reservoir simu-
lators (Thiele et al. 1997). All wells are assumed to start
producing at the same time with the same pressure control.
3 Results and discussion
Field-level and well-level simulation results are presented
and discussed to examine the impact of global and local
similarities. The relation between NGR and oil production,
Fig. 6 Workflow used to generate static models (realizations)
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Fig. 7 Distribution of porosity and permeability for each category and their relationship
Fig. 8 Three-dimensional view of resulting a permeability, b porosity for the reference model (realization), and upscaled results of
c permeability and d porosity
278 Petroleum Science (2018) 15:270–288
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and the impact of model parameters on NGR and produc-
tion values are discussed.
3.1 Controlling factors and net-to-gross ratio
A regression study was carried out to illustrate the relation
between controlling geological factors and NGR. The NGR
for each realization has been estimated for the aggregate of
all cases (Fig. 10a) and the data generated using each
equation separately (Fig. 10b) using multiple linear
regression. The statistical significance of the regression is
represented by its p value, which is the probability of being
extreme considering the null hypothesis (Utts and Heckard
2011). The coefficients of the regression are provided in
Table 5. The overall regression for the aggregate scenario
has a coefficient of determination of 0.71. Channel depths
are statistically significant (P = 8 9 10-11), while avulsion
periodicity is not significant (P = 0.42). The coefficient of
determination for each dataset, considered separately, is
larger than 0.94, and the statistical significance of avulsion
periodicity decreases as the width-to-depth ratio increases.
The results indicate that the depth-to-width ratio influ-
ences the ultimate NGR significantly (wider channels with
the same depth are more likely to deposit larger sandbod-
ies), see Fig. 11. Cumulative production for all models is
shown in Fig. 12. Compared to equations proposed by
Leeder, and Bridge and Tye, the use of Williams’s equation
(Table 2) results in larger channels and more oil produc-
tion. A higher avulsion periodicity leads to a longer
channel residence time in its bed. As a result, point bars are
laterally more extended and avulsion controls the archi-
tecture of fluvial reservoirs as point bars are the main
preserved sandbodies in meandering fluvial systems. In
such cases, the number of individual channels decreases
and a higher Av does not always result in higher cumula-
tive production. Among all cases with the same fluvial
channel geometry, the majority of the realizations with an
Av value of 40,000 years (the average of Av values of all
cases; shown in red) result in higher cumulative oil pro-
duction values, see Fig. 12. The impact of local hetero-
geneity is shown in Fig. 13, where the rate of oil
production in the most productive well (W05) is at least
one order of magnitude higher than the rate of the least
productive well (W16). Figure 13 shows a distinct differ-
ence in production from one well to another; moving
eastward from W05 to W09, total production drops con-
siderably (from 8600 to 1300 m3). This rapid change in
production indicates that productive rocks are distributed
heterogeneously in the reservoir with a scale less than that
of the well spacing.
Fig. 9 a Relative permeability curves and b capillary pressure curves for the three facies
Table 4 Reservoir and fluid properties
Properties Value Unit
Model size 2340 9 2340 9 74 m
Grid size 60 9 60 9 2 m
Reservoir thickness 74 m
Number of layers 37 –
Top of reservoir 1926 m
Initial reservoir pressure 22 MPa
Kz/Kx 0.1 –
Density (oil, water, and gas) 720.8, 1009.5, 1.12 kg/m3
Rock compressibility 4.35 1/MPa
Primary production period 450 days
Minimum economic oil rate 10 bbl
Bottom hole pressure 19.3 MPa
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The interactions of the geometric factors with avulsion
frequency are not statistically significant (as shown in
Table 5). Although Bryant et al. (1995) have shown that
avulsion rate is closely correlated with sediment feed rate,
it is rather impractical to describe the interaction between
geometric factors and avulsion quantitatively. Similar to
the approach used by Comunian et al. (2014), geometric
factors and avulsion are assumed to be independent to
simplify the analysis. A higher avulsion periodicity does
not necessarily lead to a higher NGR and thus a higher
cumulative production.
Fig. 10 Comparison between estimated NGRs and real NGRs. NGRs
are estimated based on fluvial channels’ depth and their regional
avulsion periodicity. Estimation of NGR using multivariate linear
regression was performed a using all data and b separately for each
group. Dashed line represents the ideal estimator, and the number on
a represents the equation number
Fig. 11 Relation between NGR and fluvial channel’s depth-to-width ratio—color has been used to show cases based on a single equation. As can
be seen in this figure, the variation caused by the geometry of channels (D/W) is more important than by the avulsion rate
Table 5 Coefficients of regression for NGR using channel depth and
avulsion rate for all realizations
Variables Eq. 1 Eq. 2 Eq. 3 All
Channel depth 1.04E201 1.41E201 1.13E201 1.19E201
Avulsion rate 4.70E207 3.30E-08 2.37E-07 2.47E-07
Bold values are statistically significant (a = 0.05)
280 Petroleum Science (2018) 15:270–288
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Fig. 12 Cumulative oil production curves; a realizations based on Eq. 1, b realizations based on Eq. 2, and c realizations based on Eq. 3
Fig. 13 Oil production rate from sixteen wells for the reference model (realization)
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3.2 Similarity of field-level cumulativeproduction
In the previous section, we analyzed the relation between
cumulative oil production and Hmax, Av, and the equation
that describes the geometry of the fluvial channels. In this
section, that relation is investigated using the absolute
difference in the cumulative production of the realizations
to define a distance metric. As shown in Fig. 14, entries on
the main diagonal of each distance matrix are equal to zero.
As the distance between realizations increases (dissimi-
larity matrix entries that are further away from its diago-
nal), the difference between the cumulative production of
those realizations increases as well. The production dis-
tance matrix for realizations based on Eq 1 (Bridge and
Tye—Table 1) exhibits gradual changes near the main
diagonal. The distance matrix for realizations based on
Eq 2 (Leeder, Table 1) appears to be controlled mainly by
the bankfull maximum, Hmax, whereas in the first group
(Eq 1), avulsion rate is also an important factor. The dis-
tance matrix for realizations based on Eq 3 (Williams,
Table 1) exhibits characteristics between those from the
first (Eq 1) and second (Eq 2) matrices. The average
cumulative oil production from models with an Av of
10,000 years is approximately 44,800 cubic meter (m3),
while the average cumulative oil production from models
with an Av of 40,000 years (around 46,400 m3) is closer to
the results from models with an Av of 70,000 years
(48,900 m3). Therefore, production and avulsion frequency
do not appear to have a linear relation (see Table 6). A
three-way analysis of the variance of the fields’ production
values suggests that the most important factor is the max-
imum depth of the fluvial channels followed by the equa-
tion used to estimate channel widths and avulsion
frequencies (Table 7). While the interaction between the
maximum depth of fluvial channels and the equation that is
used to calculate their widths is statistically significant, the
interactions of the avulsion rate with these parameters are
not statistically significant.
3.3 Similarity of well-level productionacross realizations
Differences between well-level production values in each
field are mainly due to local variations in geological fea-
tures in the reservoir. Figure 14 shows the comparison of
the patterns of dissimilarity of the realizations at a well
level. The dissimilarity is calculated based on the mean
root-square distance (MRSD) of well production rates
through all timesteps. The realizations based on Eq 1 show
gradual changes in the MRSDs of their production rates.
This gradual change is observed in the two other groups as
well. There is a sharp change in the field-level rate of
production between realizations with Hmax equal to 3.7 m
Fig. 14 Dissimilarity distances of realizations at a well level and
b field level using the mean root-square distance (MRSD) in their rate
of oil production along with all timesteps in three classes according to
the equation used to estimate the width of the fluvial channel;
c Scatter plot of dissimilarity distance of the fields at field level and
well level
282 Petroleum Science (2018) 15:270–288
123
(see Fig. 15). Scatter plots of the realizations’ similarity at
field and well levels are provided in Fig. 14c. These graphs
indicate a general trend between the similarities due to the
conditioning and also show a large range of variation
around the general trend line. The significance of this trend
decreases from realizations based on Eq. 1 to those based
on Eq. 2 and Eq. 3 (R2Eq1
¼ 0:58; R2Eq2
¼ 0:68;
R2Eq3
¼ 0:81). Sand deposits with a large width-to-depth
ratio (sheet-like sandbodies) tend to be more connected in a
realization, leading to less variable well production values
in that realization. The same logic may not be valid for
comparing well-level production across realizations where
impactful geological parameters for each realization, such
as NGR, are significantly different. Figure 15 shows the
cumulative production for all 16 producers in the reference
model during primary production. A given well’s produc-
tion may change significantly from one realization to the
Fig. 15 Comparison between cumulative oil productions of wells by varying the type of equation describing the width of the fluvial channel for a
high-NGR case
Table 6 Sensitivity of
cumulative production to the
main factors
Factor Factor’s Average of cumulative production, m3
Lower band Upper band Lower band All cases Upper band
Avulsion, years 10,000 70,000 44,843 47,565 48,555
Equation Eq. 1 Eq. 2 36,980 47,565 60,122
Hmax, m 2 4 12,228 47,565 70,639
Table 7 Factorial analysis of variance of statistically significant
interactions on realizations’ production
Source df Mean Sq. F Prob[F
Hmax 5 3.41 9 1014 677.19 1.4 9 10-21
Equation 2 1.78 9 1014 353.89 2.5 9 10-16
Avulsion 2 7.21 9 1012 14.32 1.4 9 10-4
Hmax 9 equation 10 3.82 9 1012 7.58 6.8 9 10-5
Hmax 9 avulsion 10 5.93 9 1011 1.18 0.361
Equation 9 avulsion 4 1.11 9 1012 2.20 0.105
df is degree of freedom; Mean Sq. represents the mean square of
errors; F is Fisher’s criteria; and Prob[F shows the probability of
the null hypothesis. Bold numbers in probability column are higher
than 0.05, which is the rejection criterion
Petroleum Science (2018) 15:270–288 283
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next, due to variations in the productive rocks’ proportion
in each parcel (80 acres around each well).
3.4 Net-to-gross ratio and cumulativeproduction
The net-to-gross ratio provides an approximation of the
reservoir quality and might be an indicator of the produc-
tive geobodies’ connectivity. Several authors have studied
the connectivity of sandbodies by varying the NGR, e.g.,
uniformly distributed sand packets (Allen 1979), Boolean
models and the truncated Gaussian method (Allard 1993),
and the object-based simulation method (Pranter and
Sommer 2011). In the present study, NGR is defined as the
volumetric proportion of sandbodies (channel lags, point
bars, and sand plugs and crevasses) to the total volume of
the reservoir. According to this study, the cumulative
production of each realization during primary production is
correlated strongly with its overall NGR (Fig. 16). Even
though one might expect field-level and well-level corre-
lations between NGR and production to be equally as
strong, it appears that well-level correlation between NGR
and cumulative production may not be as strong. Reservoir
architecture dictates the connectivity of productive zones to
each well (Fig. 17a). Therefore, architecture plays an
important role in defining the production domain of each
well. In such cases, regular compartmentalization of the
reservoir gives an incorrect estimation of the volume of the
productive rocks connected to each well.
Geobody connectivity metrics, described and discussed
in Renard and Allard (2013), are helpful in assessing
reservoir quality. The connectedness of sands to the wells
is defined based on the minimum time-of-flight from each
block to wells. The minimum time-of-flight (Datta-Gupta
and King 1995) from each well to each block is calculated
using Dijkstra’s algorithm diffusing and updating the time-
of-flight iteratively (Hetland 2014). The travel time through
each block is assumed to be proportional to the inverse of
the permeability (Fig. 17e). Figure 17b indicates that this
metric gives a better approximation. As shown in Fig. 17,
cumulative oil production is not correlated with NGR.
Even though cumulative oil production appears to be
somewhat correlated with sandbody connectedness, this
correlation does not appear to be strong enough to be used
as a reliable estimator. Since sandbodies are more channel-
like rather than sheet-like, their lateral extension impacts
their connectivity to the wells. In fact, this weak correlation
emphasizes the role of preferential paths and connectivity
of sandbodies. As shown in Fig. 17, some sandbodies are
not connected to their closest wells.
3.5 Similarity visualization
As expected, Hmax controls the overall NGR. The similarity
of the realizations is discussed in Sects. 3.3 and 3.4 based
on field- and well-level cumulative production values. The
difference in the ultimate cumulative production values
may also be used to visualize the similarity between the
realizations by projecting them in a metric space using a
multi-dimensional scaling technique. This metric space is
projected to a higher-dimensional space to increase the
models’ separability using a radial basis function kernel.
This technique has been described in detail in Caers
(2011). A 2D projection of realizations in kernel space is
Fig. 16 Relation between overall NGR and total amount of produced oil for reference cases using: a Bridge and Tye’s equation, b Williams’s
equation, c Leeder’s equation, and d all three cases
284 Petroleum Science (2018) 15:270–288
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provided in Fig. 18. The coordinates do not have a physical
meaning, but the similarity patterns are highlighted. In
Fig. 18, there is a tendency for realizations with the same
channel depth to be near each other. Proximity of the
realizations is driven most prominently by channel depth,
and then equations are used to relate width to depth, and
finally avulsion rate. A critical task in modeling mean-
dering reservoirs is the estimation of the depth of fluvial
Fig. 17 a Relation between local NGR and wells’ total amount of the
produced oil (base case according to Williams’s equation); b relation
between the proportion of the sandbodies connected to each well and
the total amount of produced oil; c streamline at 60 days; d at
240 days; e sandbodies connected to wells
Petroleum Science (2018) 15:270–288 285
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channels based on the thickness of sediment sandbodies. It
is necessary to distinguish one-story from multi-story
channels and to have an accurate model of sequence
stratigraphy for each sedimented feature to differentiate
truncated features.
4 Conclusions
This work is concerned with the geological uncertainty in
fluvial reservoirs. It examines how cumulative oil pro-
duction is impacted by the main factors that control the
architecture of fluvial reservoirs generated by meandering
systems.
Simulation results show that reservoir NGR and geo-
metrical parameters of channels impact cumulative pro-
duction more significantly than avulsion periodicity. Since
channel width is a function of the depth of the channel,
depth is the most important factor. According to the results
of global sensitivity analysis, the estimation of the width of
fluvial channels is more important than the avulsion
periodicity.
The net-to-gross ratio appears to be a satisfactory metric
to evaluate the production potential of fluvial reservoirs,
but as shown here, this metric works well only for esti-
mation of production potential at the field level and is
inadequate to estimate the well-level production. A better
metric to approximate well-level oil production is the
portion of the connected sands to each well instead of their
proportion at the vicinity of each well. Cumulative oil
production is not correlated with NGR. Even though
cumulative oil production appears to be somewhat
correlated with sandbody connectedness, this correlation
does not appear to be strong enough to be used as a reliable
estimator.
Acknowledgements The authors thank StreamSim Technologies,
FLUMY group, and Schlumberger for providing licenses for the
softwares used in this work. The authors are thankful to anonymous
reviewers for providing constructive comments.
Open Access This article is distributed under the terms of the Creative
Commons Attribution 4.0 International License (http://creative
commons.org/licenses/by/4.0/), which permits unrestricted use, dis-
tribution, and reproduction in any medium, provided you give
appropriate credit to the original author(s) and the source, provide a
link to the Creative Commons license, and indicate if changes were
made.
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